# Evaluating values of the Weierstrass $\wp$-function

I would like to know how can we evaluate the Weierstrass $\wp$-functions. That is, I would like to find $\wp(\theta,\omega,i\omega)$ for some $\theta,\omega\in\mathbb{R}$.

I'm only able to find a function which outputs the Laurent series of the Weierstrass $\wp$-function when an elliptic curve has been entered. Should I evaluate that laurent series?

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The function weierstrass_p() of EllipticCurve returns the Laurent expansion of $\wp$ at the origin, hence evaluating it only gives a reasonable approximation near it.

Nothing comes to my mind to do this kind of numerical evaluation straightforwardly in Sage. If you have access to Maple, you could give a shot at the NumGFun package, part of the AlgoLib library http://algo.inria.fr/libraries/. It has support for the numerical evaluation of functions satisfying linear differential equations with polynomial coefficients.

The author of NumGFun gave a talk on it at Sage Days 49 http://www.marc.mezzarobba.net/#expose-sd49. As you can read in the slides, nothing of it is already in Sage, but there are plans for the close future. See, for example, http://trac.sagemath.org/ticket/14996, which will add functionality similar to what you need.

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The Jacobi elliptic functions in #14996 are already in Sage, the patch just improves them.

( 2013-08-08 02:45:41 +0200 )edit

Nice, I didn't know that. Then you may try using the formulas given here: <http://en.wikipedia.org/wiki/Weierstrass%27s_elliptic_functions#Relation_to_Jacobi_elliptic_functions></p<>>

( 2013-08-09 12:04:27 +0200 )edit